Non-singular "Gauss'' black hole from non-locality

Cutting out an infinite tube around $r=0$ formally removes the Schwarzschild singularity, but without a physical mechanism this procedure seems ad hoc and artificial. In this paper we provide justification for such a mechanism by means of non-locality. Motivated by the Gauss law we define a suitable radius variable as the inverse of a regular non-local potential, and use this variable to model a non-singular black hole. The resulting geometry has a de\,Sitter core, and for generic values of the regulator there is \emph{no inner horizon}, saving this model from potential issues via mass inflation. An \emph{outer} horizon only exists for masses above a critical threshold, thereby reproducing the conjectured ``mass gap'' for black holes in non-local theories. The geometry's density and pressure terms decrease exponentially, thereby rendering it an almost-exact vacuum solution of the Einstein equations outside of astrophysical black holes. Its thermodynamic properties resemble that of the Hayward black hole, with the notable exception that for critical mass the horizon radius is zero.